In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), a central theme is to solve some fixed-point-equation on an appropriate space of probabilities, we call such an equation a Recursive Distributional Equation (RDE). Exploiting the natural recursive structure one can associate with a solution of a RDE a tree index random process which we call a Recursive Tree Process (RTP). In some sense if a RDE has a solution then the corresponding RTP is an almost sure representation of it. In this talk we will discuss several examples where such process arise naturally. We will outline some basic general theory with the main objective of determining possible influence of the boundary at infinity on the root for a recursive tree process. We will explore two aspects, the question on endogeny: the RTP being measurable with respect to the associated innovation process (the data available from the RDE), and the question of having a non-trivial tail for the RTP. We will give necessary and sufficient conditions for endogeny and tail-triviality, and will indicate some non-trivial applications of this theory. (Some part of this talk is based on a joint work with Professor David J. Aldous).